There are no complete openings refuted by computer analysis from scratch as far as I know. There has been computer assistance in the analysis of more than a few gambits, here you can look for more details. However, in modern opening theory, computer evaluations (running on powerful software) are given a very high regard by the top players, and this has caused some openings to fall in use, as being thought to not provide any advantage. One of these opening is the Italian game:
[Title "Italian game - drawing line"]
[FEN ""]
1. e4 e5 2. Nf3 Nc6 3. Bc4 Bc5 4. c3 Nf6 5. d4 exd4 6. cxd4 Bb4 7. Bd2 Nxe4 {This was considered to be a greedy line that lead nowhere for black, since white gets active play and centralizes his pieces effortlessly. Nowadays, this has become a drawing line for black.} 8. Bxb4 Nxb4 9. Bxf7 Kxf7 10. Qb3 d5 11. Ne5 Ke6! {This was the discovery of the computer.} (11...Ke8 12.Qxb4 {This was the old line, with a very comfortable game for white.}) 12. Qxb4 Qf8! {And now the black queen attacks both the white queen and the f2 pawn, forcing the exchange.} 13. Qxf8 Rxf8 {The position is absolutely equal.}
The problem is the huge number of variations the computer has to calculate. This number of different chess positions is considered to be around 10^50, and these positions can be achieved via many different move orders. Thus, the current algorithms that involve MinMax and similar weight-based methods can hardly cover a fraction of those variants.
An enlightening example is to see how the endgame tablebases are generated. They start from the mated position and go backwards, they do not use the tree-search that characterizes computer engines because it has too many branches for an efficient computation. This is a reason why a computer sometimes gives advantages in theoretically drawn positions: he does not see the end of the variation.
IMHO, it doesn't really make sense to have a project to solve an opening (at least not operating in polynomial time) due to the enormous number of possible lines and the time that takes to analyze them all.